Invertible elements of alternative unital ring form Moufang loop

Statement

Suppose $R$ is an alternative unital ring with multiplication $\!*$. Suppose $S$ is the subset of $R$ comprising those elements of $R$ that possess two-sided inverses for $\!*$. Then, $S$ is closed under $*$ and acquires the structure of a Moufang loop (?) under $*$.

Related facts

• Artin's theorem on alternative rings which states that for a ring, being alternative is the same as being diassociative.
• Alternative ring satisfies Moufang identities

Facts used

1. Alternative ring satisfies Moufang identities

Proof

The proof follows directly from fact (1).